Flow of Maxwell Fluid with Heat Transfer through Porous Medium with Thermophoresis Particle Deposition and Soret–Dufour Effects: Numerical Solution

: In this paper, we study the magnetohydrodynamics of Darcy ﬂow in a non-Newtonian liquid. The inﬂuence of thermophoresis on particle deposition is examined in the Darcy ﬂow of a Maxwell nanoﬂuid. In our model, the temperature distribution is generated by the Fourier law of heat conduction with nonlinear thermal radiation and heat sink/source. We also examine the Soret– Dufour effects in the mass concentration equations. The Brownian and thermophoretic diffusions are assumed to be generated by nanoparticle dispersion in the ﬂuid. The similarity method is used to transform the partial differential equations into nonlinear ordinary differential equations. The transformed ﬂow equations were solved numerically using the BVP Midrich scheme. The results of the computation are displayed graphically and in tabular form. The results obtained show that increasing the Deborah number leads to a decline in radial and angular motion and a decrease in the magnitude of axial ﬂow. As expected, the strength of the heat source and the values of the thermal radiation parameters determine the temperature of the liquid. We also found that as the Soret number rises (or the Dufour number falls), so does the mass transfer rate.


Introduction
Using a range of computational, numerical and theoretical approaches, [1][2][3] we investigated fluid flow with a perspective of engineering applications. Understanding the flow characteristics of non-Newtonian liquids is needed in numerous applications. For such liquids, the Navier-Stokes theory is inadequate, and there is no single constitutive equation that reflects all fluid features in the literature. As a result, a number of non-Newtonian liquid models have been proposed. Among these models, the one that has gained the most interest is the Maxwell fluid model. This model is a simplified subdivision of the rate-type of non-Newtonian fluids that allows for an authentic approximation of the phenomenon being studied. This model is named viscoelastic fluid due to its elasticity and viscosity effects and was initially proposed by James Clerk Maxwell in 1867. The Maxwell model was modeled for the purpose of predicting the elastic and viscous behavior of air [4]. However, some researchers have repeatedly applied the Maxwell model to the response of various viscoelastic fluids, ranging from polymeric fluids to the Earth's mantle. The Maxwell fluid's rheological properties were discussed by Olsson et al. [5]. Choi et al. [6] examined the Maxwell fluid flow behavior within a channel. Maxwell fluid models were investigated by Fetecau et al. [7,8], who found new analytical solutions. Mustafa [9] investigated the features of the modified Fourier law on non-Newtonian liquid using a Maxwell fluid model. The convective and radiative Maxwell fluid flow is studied by Mabood et al. [10]. The chemically reactive flow of Maxwell liquid along with heat source and Joule heating was examined by Hosseinzadeh et al. [11]. Ijaz and Ayub [12] examined the stratified flow of Maxwell nanofluid in the presence of activation energy. In [13], Mabood et al. examine the Maxwell fluid model with variable thermal conductivity. The latter paper considered the swirling motion of the liquid in a rotating disk. Recently, Devi and Maboob [14] discussed the swirling flow with heat and mass transfer featuring the relaxation time effects and obtained a numerical solution using the shooting technique. Several researchers have focused on Maxwell fluid flow analysis with various physical aspects applicable to a variety of situations [15][16][17][18][19][20][21][22][23][24].
The novel aspect of this paper lies in examining the rotation and Darcy flow of non-Newtonian fluid caused by a rotating disk; a topic that has received little attention in the past. The heat and mass transport are analyzed along with the features of the Dufour and Soret effects. The flow analysis took into account thermal radiation, heat source/sink and Joule heating effects. The conversion of the governing equations into nonlinear ordinary differential equations is carried out using the von Kármán similarity procedure. The problem is numerically integrated using the Maple BVP Midrich package. A comparison of the outcome of our computations with previously reported work is tabulated.

Formulation
In this study, we examine the steady incompressible chemically reactive Darcy flow of a Maxwell fluid influenced by Joule heating. Cylindrical coordinates (r,ϕ,z) are used in the mathematical modeling of the physical problem. We modeled the motion of the fluid as a disk rotating about the z-axis with uniform angular velocity. The disk is porous with mass flux velocity w 0 (w 0 > 0 for injection and w 0 < 0 for suction). A uniform beam of magnetic field B 0 is imposed along the z-axis. The thermophoresis effect is used to better model the fluctuation of mass deposition on the surface. The flow is axisymmetric along the z-axis. The heat and mass transport are analyzed along with the Dufour-Soret effects. The flow mechanism is shown in Figure 1.
Maxwell model to the response of various viscoelastic fluids, ranging from polymeric fluids to the Earth's mantle. The Maxwell fluid's rheological properties were discussed by Olsson et al. [5]. Choi et al. [6] examined the Maxwell fluid flow behavior within a channel. Maxwell fluid models were investigated by Fetecau et al. [7,8], who found new analytical solutions. Mustafa [9] investigated the features of the modified Fourier law on non-Newtonian liquid using a Maxwell fluid model. The convective and radiative Maxwell fluid flow is studied by Mabood et al. [10]. The chemically reactive flow of Maxwell liquid along with heat source and Joule heating was examined by Hosseinzadeh et al. [11]. Ijaz and Ayub [12] examined the stratified flow of Maxwell nanofluid in the presence of activation energy. In [13], Mabood et al. examine the Maxwell fluid model with variable thermal conductivity. The latter paper considered the swirling motion of the liquid in a rotating disk. Recently, Devi and Maboob [14] discussed the swirling flow with heat and mass transfer featuring the relaxation time effects and obtained a numerical solution using the shooting technique. Several researchers have focused on Maxwell fluid flow analysis with various physical aspects applicable to a variety of situations [15][16][17][18][19][20][21][22][23][24].
The novel aspect of this paper lies in examining the rotation and Darcy flow of non-Newtonian fluid caused by a rotating disk; a topic that has received little attention in the past. The heat and mass transport are analyzed along with the features of the Dufour and Soret effects. The flow analysis took into account thermal radiation, heat source/sink and Joule heating effects. The conversion of the governing equations into nonlinear ordinary differential equations is carried out using the von Kármán similarity procedure. The problem is numerically integrated using the Maple BVP Midrich package. A comparison of the outcome of our computations with previously reported work is tabulated.

Formulation
In this study, we examine the steady incompressible chemically reactive Darcy flow of a Maxwell fluid influenced by Joule heating. Cylindrical coordinates (r,φ,z) are used in the mathematical modeling of the physical problem. We modeled the motion of the fluid as a disk rotating about the -axis with uniform angular velocity. The disk is porous with mass flux velocity ( > 0 for injection and < 0 for suction). A uniform beam of magnetic field is imposed along the z-axis. The thermophoresis effect is used to better model the fluctuation of mass deposition on the surface. The flow is axisymmetric along the -axis. The heat and mass transport are analyzed along with the Dufour-Soret effects. The flow mechanism is shown in Figure 1.  Our assumptions, given above, result in the following governing equations: ∂u ∂r The radiative flux q rad is given by The thermophoretic velocities are where k is the thermophoretic coefficient whose value is usually taken to fall in the range of 0.2-1.2 and, according to Batchelor and Shen [26] and Talbot et al. [27], is given by Using the similarity variables [28], Substituting Equation (10) into Equations (1)- (5), we obtain 1 The transformed BCs are In (16), prime denotes differentiation with respect to η. The parameters are expressed as Here, M is the magnetic field parameter, R the stretching parameter, β the Deborah number, γ the porosity parameter, s the mass transfer parameter, Rd the radiation parameter, Bi the Biot number, θ w the temperature ratio parameter, δ the heat source/sink parameter, Ec the Eckert number, Du the Dufour number, K r the chemical reaction parameter, Sr the Soret number, Pr the Prandtl number and Sc the Schmidt number. Finally, Nt is the relative temperature difference parameter, which is negative for a heated surface, positive for a cooled surface and zero for surfaces at ambient temperature.
The physical parameters are defined as: where Nu r is the Nusselt number, and Sh r is Sherwood number. Their dimensionless forms are in which Re = r 2 Ω ν is the local Reynold number.

Results and Discussion
In this section, we examine the effects of the above parameters on the flow fields, temperature and concentration distributions. To do this, the governing Equations (11)(12)(13)(14)(15) and conditions (16) were numerically solved using the Maple BVP function with Midrich method. The results are shown by graphing all the governing parameters, namely the magnetic field parameters, the relaxation time parameter, the Soret number, the porosity parameter, the Dufour parameter, the Biot number, the Prandtl number, the chemical reaction parameter, the suction parameter and the Schmidt number against the fluid's velocities, temperature and concentration distributions, as shown in  sionless variable . In an electrically conducting fluid, a drag force, which is essentially a resistive force, is produced by the magnetic field. This resistive force tends to reduce the movement of fluid and increase the temperature on the surface of the disk. This explains the decrease in the values of radial , azimuthal and the magnitude of the axial velocity seen in Figures 2a-c. The increase in the temperature and mass concentration of the fluid is shown in Figures 2 d,e.    (F, G, H). The effect of the Deborah number on temperature distribution is displayed in Figure 3d, where it can be seen that the temperature decreases with the increasing β. rah number on temperature distribution is displayed in Figure 3d, where it can be seen that the temperature decreases with the increasing . An increase in the thermal radiation parameter, as can be expected, results in an increase in the temperature of the liquid, as seen in Figure 4a. To maintain the heat transportation in the liquid, heat source/sink effects are applied. As seen in Figure 4b, an upward shift in the temperature profile is shown as the values of increase from 0.0 to 0.3. An increase in the thermal radiation parameter, as can be expected, results in an increase in the temperature of the liquid, as seen in Figure 4a. To maintain the heat transportation in the liquid, heat source/sink effects are applied. As seen in Figure 4b, an upward shift in the temperature profile is shown as the values of δ increase from 0.0 to 0.3.  The influence of the Dufour number and the Soret number on the temperature distribution is shown in Figure 5a. In this figure, one sees an upward shift of the thermal curves with increasing . Figure 5b shows the effect of the Dufour number and the Soret number on the solutal concentration. Increasing the Soret , while decreasing the Dufour number , results in an upward shift of mass concentration. Figures 6 a,b show that the solutal concentration increases by increasing the values of The influence of the Dufour number Du and the Soret number Sr on the temperature distribution is shown in Figure 5a. In this figure, one sees an upward shift of the thermal curves θ(η) with increasing Sr. Figure 5b shows the effect of the Dufour number Du and the Soret number Sr on the solutal concentration. Increasing the Soret Sr, while decreasing the Dufour number Du, results in an upward shift of mass concentration. Figure 6a,  The influence of the Dufour number and the Soret number on the temperature distribution is shown in Figure 5a. In this figure, one sees an upward shift of the thermal curves with increasing . Figure 5b shows the effect of the Dufour number and the Soret number on the solutal concentration. Increasing the Soret , while decreasing the Dufour number , results in an upward shift of mass concentration. Figures 6 a,b show that the solutal concentration increases by increasing the values of and .   and . Increasing the values of , and , the heat transfer rate rises while the effect of and are to reduce the heat transfer rate in the liquid. The tabular results of the Sherwood number against , , and are shown in Table 2. As can be seen in Table 2, increasing the values , , and decreases the mass transfer rate. Finally, Table  3 shows the relationship between the results obtained in this paper and past results [29,30].   Increasing the values of Rd, Bi and Du, the heat transfer rate rises while the effect of Sr and Ec are to reduce the heat transfer rate in the liquid. The tabular results of the Sherwood number against γ, K r , k and Nt are shown in Table 2. As can be seen in Table 2, increasing the values γ, K r , k and Nt decreases the mass transfer rate. Finally, Table 3 shows the relationship between the results obtained in this paper and past results [29,30].  Table 3. The relationship between [29,30] and the current paper on fixed Pr = 6.5 and M = 0 = γ = R = s = β = Rd = δ = Ec = Du.